Errors in the OLAP storage engine: The metadata for the statically linked measure group, with the name of xxxxx, cannot be verified against the source object
Luckily Google came to the rescue with this MSDN forum thread, the jist of which is that if you add a linked measure group to a cube, the BIDS interface will let you delete some of the measures in the group (since you may not want to show all the available measures in your new cube) - however, doing this causes the above error when processing the cube! The error is basically saying that the linked measure group metadata no longer matches that of the source measure group.
The solution is that when you add a linked measure group, all the existing measures should be left intact. To hide them from the end user, set the "Visible" property for each measure.